Fully removing other AV solutions is correct way to go as described in this thread.
Being connected to the internet should not affect the security of the installation.
If you are using older AVG product and you are upgrading to AVG 2012 you can choose not to uninstall prior to AVG 2012 installation in order to preserve your configuration.
Sometimes there is different opinions whether a program needs to be uninstalled before adding a new version. I just removed the existing anti virus program on our other computer (VISTA) A different virus program than AVG. so in that instance uninstall makes sense. I downloaded the new AVG to desktop first. Then disconnected from the internet connection (for safety) before uninstalling the old program. Once finished with that I started the installation of AVG. When it was ready to update, I connected my computer back to cable modem. Right way to go?